Now we develop the splitting rules. Many random survival forest methods use the two-sample log-rank test statistic to partition the population into subgroups with a homogeneous survival distribution [1]}, [2]}. When failure time is not observed as a scalar but is given as a probabilistic process, the log-rank test can be generalized to the generalized log-rank statistic [3]};
\(LR(\mathcal {N}_1, \mathcal {N}_2) = \frac{\int _0^\tau \frac{Y_2(t){dN_1(t)}+ Y_1(t){dN_2(t)}}{Y(t)}}{\sqrt{\int _0^\tau \frac{Y_1(t)Y_2(t)dN(t)( Y(t)- dN(t))}{Y(t)^3}}},\) 
